Drinfeld realization of the elliptic Hall algebra

By O. Schiffmann. 2010

72 citations recorded by the source corpus.

We give a new presentation of the Drinfeld double $\boldsymbol{\mathcal {E}}$ of the (spherical) elliptic Hall algebra $\boldsymbol{\mathcal{E}}^{+}$ introduced in our previous work (Burban and Schiffmann in Duke Math. J. preprint math.AG/0505148, 2005). This presentation is similar in spirit to Drinfeld’s ‘new realization’ of quantum affine algebras. This answers, in the case of elliptic curves, a question of Kapranov concerning functional relations satisfied by (principal, unramified) Eisenstein series for GL(n) over a function field. It also provides proofs of some recent conjectures of Feigin, Feigin, Jimbo, Miwa and Mukhin (arXiv:1002.3100, 2010).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2ddf70cfcd6e0de38e3e28b4953dd25981a338dc

How researchers see this paper

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In Geometry and Topology

1. A. Tsymbaliuk
2. Boris Feigin
3. O. Schiffmann
4. Sabin Cautis
5. Anthony M. Licata

In Mathematical Physics

1. Boris Feigin
2. A. Tsymbaliuk

Citations in the corpus

• Quantum Geometry and Quiver Gauge Theories (2013, unknown): “It was called quantum continuous gl∞, or deformation of universal enveloping of qdifference operators in one-variable in [177, 178]; spherical elliptic Hall algebra or gl∞ DAHA in [179] and [165, 180].”
• Equivariant K-theory of Hilbert schemes via shuffle algebra (2009, background): “In fact, the description of the ideal Ie is conjectured in [2] and is recently proved in [9].”
• The affine Yangian of gl1 revisited (2014, unknown): “[18] The assignment u1,i → ei, u−1,i → fi, θ0,±j → ψ±j · ψ∓1 0 , κa,b → ψ 0 for i, a, b ∈ Z, j ∈ N extends to an isomorphism of algebras Ξ : Ẽ/(κ0,1 − 1) ∼ −→Üq1,q2,q3(gl1).”